📚 Mastering Quadratic Equations and Graphs for IGCSE Maths | IGCSE 数学二次方程与图像完全攻略
Quadratic equations and their graphs form a central topic in the IGCSE Mathematics syllabus. Whether you are sitting the Core or Extended paper, you are expected to solve quadratics, interpret parabolas, and apply the discriminant. This revision guide brings together all the key methods and exam strategies in one bilingual resource.
二次方程及其图像是 IGCSE 数学课程大纲中的核心主题。无论你参加 Core 还是 Extended 考试，都需要掌握求解二次方程、解读抛物线以及应用判别式。本复习指南将关键方法和考试策略整合在一份双语资源中。
1. What Is a Quadratic Expression? | 什么是二次表达式
es linear, so the condition a ≠ 0 keeps the x² term present. In IGCSE questions you will see quadratics written in different orders, such as c + bx + ax², but they can always be rearranged into standard form.
二次表达式是次数为 2 的多项式，其标准形式为 ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，表达式就退化为一次式，因此条件 a ≠ 0 保证 x² 项存在。在 IGCSE 题目中，你可能会看到二次式以不同顺序书写，例如 c + bx + ax²，但它们总可以整理成标准形式。
ax² + bx + c = 0, a ≠ 0
When a quadratic expression is set equal to zero, it becomes a quadratic equation. The solutions to this equation are called roots, and they can be real or complex. At IGCSE level, you will mainly work with real roots unless the discriminant is negative.
当二次表达式设为零时，就变成二次方程。这个方程的解称为根，根可以是实数或复数。在 IGCSE 阶段，除非判别式为负，否则你主要处理的是实根。
2. Solving by Factorising | 因式分解法求解
s usually fastest, but it only works when the expression has rational factors. Always check that the factorised form expands back to the original quadratic.
如果二次方程可以因式分解成两个一次括号，就可以利用零乘积性质求解。例如 x² + 5x + 6 = 0 可写成 (x + 2)(x + 3) = 0，因此 x + 2 = 0 或 x + 3 = 0，解得 x = -2 或 x = -3。这种方法通常最快，但只有当表达式存在有理因式时才适用。一定要检查因式展开后是否等于原二次式。
actise splitting the middle term so that you can quickly find the correct pair of numbers.
当首项系数 a 不为 1 时，因式分解需要更加仔细。例如 2x² + 7x + 3 可以写成 (2x + 1)(x + 3)，因为 2x² + 6x + x + 3 = 2x² + 7x + 3。请练习拆分中间项，以便快速找到正确的一对数。
3. Completing the Square | 配方法
- 9 + 1 = (x + 3)² - 8. When solving x² + 6x + 1 = 0, this gives (x + 3)² = 8, so x = -3 ± √8 = -3 ± 2√2.
配方法将二次式改写为 a(x + p)² + q 的形式。这种形式可以直接看出顶点，也用于推导二次公式。以 x² + 6x + 1 为例，取 6 的一半并平方得到 9，于是写成 (x + 3)² - 9 + 1 = (x + 3)² - 8。解方程 x² + 6x + 1 = 0 时，可得 (x + 3)² = 8，所以 x = -3 ± √8 = -3 ± 2√2。
x² + bx + c = (x + b/2)² − (b/2)² + c
Completing the square is especially useful when the roots are irrational or when you need to find the minimum or maximum value of a quadratic function. Make sure you keep the signs correct when a constant term is negative.
当根是无理数或需要求二次函数的最小值或最大值时，配方法特别有用。当常数项为负数时，务必保持符号正确。
4. The Quadratic Formula | 二次公式
ntire -b in the numerator. Many exam questions ask you to round answers to a given number of decimal places, so keep intermediate values in your calculator rather than rounding too early.
二次公式可以求解任意二次方程 ax² + bx + c = 0。它由配方法推导而来，公式如下。你必须准确记忆公式，包括 ± 号和分子中的整个 -b。许多考试题要求将答案四舍五入到指定小数位，因此计算过程中要保留中间值，不要过早取整。
x = (−b ± √b² − 4ac) ÷ 2a
− 2x − 1 = 0, identify a = 3, b = -2, c = -1. Substituting into the formula gives x = (2 ± √4 + 12) ÷ 6 = (2 ± √16) ÷ 6, so x = (2 ± 4) ÷ 6. The two solutions are x = 1 and x = -1/3.
例如，解方程 3x² − 2x − 1 = 0，确定 a = 3，b = -2，c = -1。代入公式得到 x = (2 ± √4 + 12) ÷ 6 = (2 ± √16) ÷ 6，所以 x = (2 ± 4) ÷ 6。两个解分别是 x = 1 和 x = -1/3。
5. The Discriminant and Nature of Roots | 判别式与根的性质
re is one repeated real root, meaning the graph touches the x-axis. If Δ < 0, there are no real roots, and the graph does not intersect the x-axis. You should be able to link these cases to the number of x-intercepts on a quadratic graph.
判别式 Δ = b² - 4ac 可以在不求解方程的情况下判断根的性质。若 Δ > 0，方程有两个不相等的实根；若 Δ = 0，有一个重根，即图像与 x 轴相切；若 Δ < 0，没有实根，图像不与 x 轴相交。你需要将这些情况与二次函数图像中 x 轴截距的数量联系起来。
| Condition | Nature of roots | Graph behaviour |
| Δ > 0 | Two distinct real roots | Two x-intercepts |
| Δ = 0 | One repeated real root | Touches x-axis at one point |
| Δ < 0 | No real roots | No x-intercepts |
In extended IGCSE questions, you may also be asked to find the set of values of a parameter for which a quadratic equation has two real roots, one root, or no roots. This requires setting up and solving an inequality based on the discriminant.
在 Extended IGCSE 题目中，还可能要求你求某个参数的值范围，使得二次方程有两个实根、一个实根或没有实根。这就需要根据判别式建立并求解不等式。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
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